On geometric variational problems in weak ambient spaces : being an account of three novel contributions in the fields of minimal surfaces and complex geometry

In this dissertation, we consider geometric variational problems in ambient spaces subject to weaker assumptions than what is considered standard. After an overview of the classical theory of harmonic maps, we explore the first of these variational problems. We prove the existence of energy-minimizing and area-minimizing ρ-equivariant maps from the universal cover of a compact Riemann surface M of genus at least one into a locally compact CAT(1) space X, where ρ: π₁(M) → Isom(X) is an isometric action of the fundamental group on X, under natural geometric assumptions on the action. This is based on joint work with Ailana Fraser. Next, we review some basic concepts from complex differential geometry, and study the geometry of Calabi—Yau conifold transitions. This deformation process is known to possibly connect a Kähler threefold to a non-Kähler threefold. We use balanced and Hermitian Yang—Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov—Hausdorff topology. This part of the dissertation is based on joint work with Sébastien Picard and Caleb Suan. Finally, we introduce the notion of a (perturbed) special Lagrangian (SL) submanifold of a (possibly non-Kähler) Calabi—Yau threefold (X,ω,Ω). Using the Sard—Smale technique, we prove the existence of a comeagre set of Hermitian metrics ω on X such that the moduli space of perturbed SL submanifolds in (X,ω,Ω) consists of isolated points.